Evaluating the channel capacity is of great interest in the design of underwater (UW) acoustic communication systems. In this paper, a parallel subchannel communication model for the acoustic UW channel is presented and a new analytic suboptimal power allocation scheme, named average-water-fllling (AWF), is proposed. The communication model is based on Monte-Carlo physical propagation simulations through an ocean environment with internal waves (IW). It is assumed that the channel state information (CSI) is not available to the transmitter. The subchannels in this model correspond to different acoustic frequency bands. The channel exhibits nonuniform frequency selective fading, where the fading intensity of each subchannel is found to be exponentially distributed. The proposed AWF scheme is based on the analysis of the water-filling (WF) power allocation, which is optimal when the transmitter is provided with the CSI. Simulations show that the AWF power allocation outperforms the uniform power allocation for any signal-to-noise ratio (SNR), and converges to the optimal performance (i.e., capacity achieving allocation) for high SNRs. As a suboptimal solution, the AWF yields an analytic lower bound for the capacity.